Root functions of a meromorphic matrix function and applications

TL;DR

This paper introduces a practical method for analyzing root and pole cancellation functions of meromorphic matrix functions, with applications to solving nonlinear differential systems and establishing a new factorization approach involving bounded operators.

Contribution

It develops a novel factorization technique for meromorphic matrix functions using root functions, extending Krein-Langer representations with bounded operators, and applies this to differential equations.

Findings

Established a factorization for meromorphic matrix functions involving bounded operators.

Demonstrated the method's application to nonlinear differential systems.

Connected the spectra of operators in the new representation with classical Krein-Langer spectra.

Abstract

A practical method is presented for determining root and pole cancellation functions of a matrix function $Q(z)$ meromorphic on the extended complex plane $\bar{\mathbb{C}}:=\mathbb{C} \cup \left\{ \infty \right\}$. This method is applied to solve a nonlinear system of $n\in \mathbb{N}$ differential equations of order $l\in \mathbb{N}$ with $n $ unknown functions $u_{i}\left( t \right)$, where $i=1,\, \mathellipsis ,\,n $. For a function $Q\in \mathcal{N}_{\kappa}(\mathcal{H}) ,\, \kappa \in \mathbb{N} \cup \lbrace 0 \rbrace$, posesing a pole at infinity of order $m \in \mathbb{N}$, the following factorization is establish \[ Q(z)=(z-\beta)^{m}\tilde{Q}(z), \, z\in \mathcal{D}(Q), \] where $\beta \in \mathbb{R}$ is a regular point of $Q$, and $\tilde{Q}\in \mathcal{N}_{\kappa'}(\mathcal{H})$ is holomotphic at $\infty$. Unlike the Krein-Langer representation of $Q$, which involves a linear relation $A$, this representation employs a bounded operator $\tilde{A}$ in the Krein-Langer representation of $\tilde{Q}$. The operator $\tilde{A}$ and the relation $A$ have identical spectra, except at $\beta$ and $\infty$. We demonstrate how to obtain this representation for a given meromorphic function $Q\in \mathcal{N}_{\kappa}^{n \times n}$ using the root functions developed in this work.